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VILA
VÍLA - VItrum LAugaricioJoint Glass Center
of Institute of Inorganic Chemistry of SAS, A. Dubček University of Trenčín, and RONA, j.s.c.
Študentská 2, Trenčín, SK-911 50, Slovak Republic
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VILA & FU AVČR & UPCE
KLASICKÁ NUKLEAČNE RASTOVÁ TEÓRIA A JEJ APLIKÁCIE
Marek Liška, Mária Chromčíková, Miroslava Rodová, Karel Nitsch, Zdeněk Černošek
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OUTLINE
• Gibbs energy controls the [P, T] world
• Driving force for crystallization
• Crystal nucleation - embryo
• Crystal growth
• Nucleation and growth – TTT
• Mathematical model
• Application to ZnCl2
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PERSISTING WARNING
NOT
TOO
MUCH
THEORY!!!!!!!!!!!!
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GIBBS ENERGY
S
S
S
L
L
L
As2Se3 - G vs T(K)FACT53 - FACT53 - FactSage 5.3 compound database (Jan. 2007)
T(K)
G(K
J/m
ol)
300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500-650
-600
-550
-500
-450
-400
-350
-300
-250
-200
-150
-100
-50
Tliq
Undercooling
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VOLUME VERSUS SURFACE23 4
34
rSrV
fpmelt,crystal, 0
)()(TT
V
TGTGG
m
mmV
0 SG
23 434
)( rGrGSGVrG VSV
2484d
)(d 2 VV GrrrGr
rrG
VV G
rGrrG 2
20max)( ***
Critical nucleus – critical radius r*:
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VOLUME
VERSUS
SURFACE
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Thermodynamic activation barrier:MISSION IMPOSSIBLE
Entropy of mixing decreases G:
N
N
N
NkTrGN
NN
N
NN
N
NN
N
NN
NkTrGNG
rrr
rrr
r
r
rr
ln)(
lnln)(
kTGN
kT
GNN r
2V
3*
*
3
16expexp
2mfpmfp
2fp
3
2V
3*
3
16
3
16
VTTH
T
GG
Vmfpfpmfpmfpmfpmfp )( GVTTTHSTHTG
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Proportional to # of neighbors, Ns, frequency, kT/h, Boltzman factor of kinetic activation energy, and Nr*:
HOMOGENEOUS NUCLEATION RATE
kTHT
TVN
kT
G
h
kTNI
2mfp
2fp
2m
3
as
3
16expexp
2fp
as
11)/ln()ln(ln
TTTB
Tk
GhkTNNI
kHTVB 2mfp
2fp
2m
3 316
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HOMOGENEOUS NUCLEATION RATE
TTT fp
kTHT
TV2
mfp
2fp
2m
3
3
16exp
kTG
hkT aexp
Decreases with T
Increases with T
Increases with T
= 0 for T=Tfp
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RATE OF CRYSTAL GROWTH
kTNGGfNdt
dnAmfpas exp)lc(
kTGfNdtdn
as exp)cl(
hkT
f
uNRTGkTGfNdt
dn
dt
dns mfpas exp1exp)lc()cl(
fp
fpmfpa exp1exp
TTR
TTHkTG
h
kTu
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NUCLEATION vs CRYSTAL GROWTH
T Tfp fpGm 0 u 0T 0 u 0
Tfp
Tg
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NUCLEATION AND GROWTHConstant p, T, single component => u, I = const.
31 34
utV Volume after growth from to t
dutIVtVt
0
3liqcr 3
4)( Vcr<<Vliq=> Vliq= const
43
0
33
liq
cr
3
1d
3
4
)0(
)()( tuItuI
tV
tVt
t
TTT = Temperature – Time – Transformation
.)0(
)()(
liq
cr consttV
tVt
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TTT DIAGRAMS
TTT diagram of salol
(A) = 10-6 (B) = 10-8
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THE MODEL
)3/ln(ln4)10ln()ln( exp6
exp3 tIu
TC
TTTBAWIu
11)ln( 2
fp
exp3
fp
2fpmfpexp1ln3)/ln(4TTR
TTHhkTW
3ss lnln3ln3ln3 NNNNA
k
GC a4
kHTVB 2
mfp2
fp2
m3 316
64expexp
3 1031
)( tuIt
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REGRESSION TREATMENT
3ss lnln3ln3ln3 NNNNA
k
GC a4
CzBxAWIuy )ln( 3
kHTVB 2mfp
2fp
2m
3 316
2fp1 TTTx
Tz 1
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RESULTS
A = - 49.11
B = 46916278
C = 46795
Ga = 97 kJ/mol
sapr = 3.4 K
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RESULTS
2 3 4 5 6 7 8 9150
200
250
300
Calculated Experiment
T /
oC
log(t / s)
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CONCLUSION
The plausible semi-empirical method (e.g. the general form of the regression
function)was proposed for the quantitative description of the TTT diagrams.
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GENERAL CONCLUSIONSSome people do not like theory...
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AT LEASTONE
EXPERIMENTPLEASE!!!
(Prayerof
MiaFarrow)
Photo
© Z. Černošek
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THANK YOU FOR YOUR PATIENT ATTENTION